Your grandma is faster than me. I have an Acer M1100 and I'm still using it.

If the original Socket is an AM2 and the replacement processor is an AM2 as well there should be no problem but sometimes the BIOS version must be updated to receive a better performing processor that is not the same as what it was intended to do. The BIOS could be complicated as I do not expect ODM like Acer will ever release a BIOS update. A 2.1Ghz dual core seems acceptable still by today's standards.

I suggest that you try to install the XFX Radeon HD 5550 first to see whether there is any improvement. If the problem has not improved then you can consider changing the processor as well. But from what I can tell a GC in that PC should solve the problems already. A clean format of the PC to Windows 7 should speed things up a little as well.

Not worth the extra money if you're going to just upgrade to another dual core processor that has a little bit better performance but with the same number of cores. I even think an Athlon II X4 635 at about the same price as the Phenom II X2 555 would be a better choice. If you want to use an AM3 processor like the Athlon II X4 635 on an AM2 socket you would need to update the BIOS.

Installing a 64bit OS on a 2GB PC is not very efficient as 64bit does take a little bit more RAM compared to a 32bit. Since you do not have more than 4GB of RAM I don't see the need for a 64bit OS. But never mind, it works anyways.

Like I said earlier, upgrade your GC first before upgrading your processor. Not necessary to upgrade both at the same time and if still notice any problems then upgrade the processor afterward.

Even an AMD Zacate with 1.6Ghz dual core + HD6310 IGP can run things pretty well I don't see why your PC would struggle with the same 2.1Ghz dual core which in fact is still far better than the AMD Zacate's processor in terms of processing power.